On Discrete-Time Output Negative Imaginary Systems

This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the z-domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter δ ≥ 0, which indicates the strictness properties. δ > 0 specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMI-based state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of δ. The paper also establishes the connections between the discrete-time Passive and discrete-time NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at z = -1 and z = +1.